reserve p,q,r for FinSequence;
reserve u,v,x,y,y1,y2,z for object, A,D,X,Y for set;
reserve i,j,k,l,m,n for Nat;

theorem
  p - {x} = p iff not x in rng p
proof
  thus p - {x} = p implies not x in rng p
  proof
    assume p - {x} = p; then
A1: {x} misses rng p by Th67;
    x in {x} by TARSKI:def 1;
    hence thesis by A1,XBOOLE_0:3;
  end;
  assume
A2: not x in rng p;
  {x} misses rng p
  proof
    assume {x} meets rng p;
    then ex y being object st y in {x} & y in rng p by XBOOLE_0:3;
    hence thesis by A2,TARSKI:def 1;
  end;
  hence thesis by Th67;
end;
